The density of a crystal

The theoretical density of a crystal can be found by calculating the mass of all the atoms in the unit 

The atomic contents of the unit cell give the composition of the material. The theoretical density of a crystal can be found by calculating the mass of all the atoms in the unit cell. (The mass of an atom is its molar mass divided by the Avogadro constant see Section Sl.l). The mass is divided by the unit cell volume. To count the number of atoms in a unit cell, we use the following information  

A quick method to count the number of atoms in a unit cell is to displace the unit cell outline to remove all atoms from corners, edges and faces. The atoms remaining, which represent the unit cell contents, are all within the boundary of the unit cell and count as 1. 

The general procedure is to determine the unit cell dimensions, the crystal structure type and the real composition of the material. The ideal composition of the unit cell will be known from the stracture type. The ideal composition is adjusted by the addition of extra atoms (interstitials or substituted atoms) or removal of atoms (vacancies) to agree with the real composition. A calculation of the density of the sample assuming either that interstitials or vacancies are present is then made. This is compared with the meastued density to discriminate between the two alternatives. 

The method can be illustrated by reference to iron monoxide. Iron monoxide, often known by its mineral name of wilstite, has the halite (NaCl) stmcture. In the normal halite structure, there are four metal and four nonmetal atoms in the unit cell, and compounds with this structure have an ideal composition MXi q (see Section 5.3.9 for further 

Model A Assume that the iron atoms in the crystal are in a perfect array, identical to the metal atoms in halite, and that an excess of oxygen is due to interstitial oxygen atoms present in addition to those on the normal anion positions. The ideal unit cell of the structure contains four iron atoms and four oxygen atoms and so, in this model, the unit cell must contain four atoms of iron and 4( 1 + x) atoms of oxygen. The unit cell contents are Fe404+4 c and the composition is FeOi.oss- 

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The presence of small concentrations of point defects changes the density of a crystal, and four values of the density can be calculated, depending on... 

An iron deficiency could be accommodated by a defect structure in two ways either iron vacancies, giving the formula Fe] /D, or alternatively, there could be an excess of oxygen in interstitial positions, with the formula FeOi+ f. A comparison of the theoretical and measured densities of the crystal distinguishes between the alternatives. The easiest method of measuring the density of a crystal is the flotation method. Liquids of differing densities which dissolve in each other, are mixed together until a mixture is found that will just suspend the crystal so that it neither floats nor sinks. The density of that liquid mixture must then be the same as that of the crystal, and it can be found by weighing an accurately measured volume. 

Keeping in mind that a crystal is composed of many of its unit cells and assuming there is no contamination, the density of a crystal can be computed from the properties of the unit cell. It is necessary to apportion the mass of the crystal among the various unit cells, and then to divide the mass apportioned to one unit cell by the volume of the unit cell. In computing the mass of a unit cell, it is important to assign to the cell only that fraction of... 

Measurement of the density of a crystal permits calculation of the weight of the contents of the unit cell. 

Frenkel and Schottky point defects in crystals defined. The effects they have on the density of a crystal... 

Thus physical properties can be used as a probe of symmetry, and can reveal the crystallographic point group of the phase. Note that Neumann s principle states that the symmetry elements of a physical property must include those present in the point group, and not that the symmetry elements are identical with those of the point group. This means that a physical property may show more symmetry elements than the point group, and so not all properties are equally useful for revealing tme point group symmetry. For example, the density of a crystal is controlled by the unit cell size and contents, but the symmetry of the material is irrelevant, (see Chapter 1). Properties similar to density, which do not reveal symmetry are called non-directional. Directional properties, on the other hand, may reveal symmetry. 

The weight of all the atoms or ions in the unit cell can be calculated if the unit cell dimensions and the density of the crystal are known. This means that crystal density measurements are important in the early stages of a crystal structure analysis. A vertical column with a density gradient, calibrated with crystals of known density, is used to measure protein densities. These are determined by where along the column the crystal setdes and so that, by interpolation, its density can be estimated (39). This method must be done with care to ensure that the measurement is good. The density of a crystal is given by... 

Will the density of a crystal go up or down if it contains (a) Schottky defects (b) Frenkel defects (c) vacancies (d) interstitials ... 

Figure 12.19 also summarizes the relationship between the atomic radius r and the edge length a of a simple cubic cell, a body-centered cubic cell, and a face-centered cubic cell. This relationship can be used to determine the density of a crystal, as Example 12.3 shows. 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

All noble gases except helium crystallize with ccp structures at very low temperatures. Find an equation relating the atomic radius to the density of a ccp solid of given molar mass and apply it to deduce the atomic radius of each of the following noble gases, given the density of each (in g-cm ) Ne, 1.20 Ar, 1.40 Kr, 2.16 Xe, 2.83 Rn, 4.4 (estimated). 

This sum over a// reciprocal space vectors of the form (IV.2) should be carefully distinguished from the expansion (III.4) of the density of a periodic crystal. If the density has the "little period", the e>mansion (IV.3) reduces to a sum over all reciprocal lattice vectors. The general case (IV.3) and the periodic case (III.4) actually represent two extreme cases. The presence of "more and more symmetry" in the density can be gauged... 

Here, generally speaking, 8- (or 8+) is a function of ev (or v+), i.e., the position of the Fermi level at the sin-face depends on its position in the interior of a crystal. In the particular case of the so-called quasi-isolated surface e9- and v (or es+ and v+) are independent parameters (1). Note that the case of a quasi-isolated surface is very widespread. It is realized when the density of surface states attains a sufficient value. 

We want to introduce the properties of the crystal and of the X-rays and solve f or the electric displacement or flux density, D. Hart gives a careful discussion of the polarisability of a crystal, showing that a sufficient model of the crystal for X-ray scattering is a Fourier sum of either the electron density or the electric susceptibility over all the reciprocal lattice vectors h. Thus the crystal is represented as... 

It is often useful to express no in terms of the volume Vz and of the density of a zeolite crystal ... 

Several formulations were proposed [65, 66], but the intuitive notation introduced by Hansen and Coppens [67] afterwards became the most popular. Within this method, the electron density of a crystal is expanded in atomic contributions. The expansion is better understood in terms of rigid pseudoatoms, i.e., atoms that behave stmcturally according to their electron charge distribution and rigidly follow the nuclear motion. A pseudoatom density is expanded according to its electronic stiucture, for simplicity reduced to the core and the valence electron densities (but in principle each atomic shell could be independently refined). Thus,... 

The crystal property perhaps most sensitive to structure is density and, conversely, the increase in density of a crystal with pressure is accompanied by significant structural changes. The latter can be of two types - either a smooth variation of the free parameters of the structure with pressure, or a first-order transition to a new structure type. 

Calculate the density of a compound from its crystal structure and atomic mass. 

The theoretical density of a crystal can be obtained from the volume of the unit cell and the mass of the unit cell contents. The results of an X-ray diffraction structure determination gives both of these data, as the unit cell dimensions are accurately measured and the type and number of formula units in the unit cell are also determined. An example of this type of calculation for FeO follows ... 

Octanitrocubane (ONC) is a white solid, somewhat soluble in hexane and readily soluble in polar organic solvents. The density of one of the ONC polymorphs is very high (1.979gcm"3) but is still lower than the calculated value (the latest and most sophisticated calculation predicts a density above 2.1 gem"3 for the most stable polymorph of ONC) which indicates the existence of a crystal form of ONC much more dense than that synthesized. Kamlet-Jacobs equations predicted that ONC is 15-30% better than HMX [109] (a most powerful currently employed military explosive) and 6% better (perhaps also less shock sensitive) than the recently discovered explosive HNIW [121, 253-258] or CL-20 as shown in Table 2.15. It is interesting to note that both HpNC and ONC have decomposition points well above 200 °C and are not detonated by hammer blows. 

The density of explosive fillings contained in munitions should be as close as possible to the theoretical maximum density (TMD) of formulations, which is calculated from the crystal densities of ingredients of an explosive formulation, taking into account their relative proportions. The density of a formulation directly affects its performance as is shown by an empirical relationship (Equation 3.1) ... 

Physical Properties.7—Tantalum is a white metal with a greyish tinge and is very similar to platinum in colour and general appearance. When it is heated to 1600° C. in vacuo it assumes a crystalline form.8 Examination of the powdered metal by X-ray analysis has shown that the arrangement of the atoms is on the plan of a body-centred cube of side 3 272 A, obtained by dividing the space of a crystal into equal closely packed cubes and placing an atom at each cube comer and each cube centre the distance between the nearest atoms is 2 883 A. The specific gravity of the fused metal is 16 6, - a sample drawn into wire 0 05 mm. diameter had a density of 16 5 10 the density calculated from X-ray data is 17 09.u... 

Crystallographers have a rough rule, which however works well over a wide range of materials, that one can assess the density of any crystal by ignoring the hydrogen atoms and allocating a volume of about 18 A3 to each atom of any other kind. For a solid of composition H2O the density should be... 

Determine the density of a metal or ionic solid from its crystal structure, Examples 5.5 and 5.7. 

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